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Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC

Simon Badger, Heribertus Bayu Hartanto, Rene Poncelet, Zihao Wu, Yang Zhang, Simone Zoia

TL;DR

The paper delivers the first full-colour analytic two-loop five-point amplitudes for $pp\to b\bar{b}H$ in the five-flavour scheme, including complete colour structures and finite Yukawa coupling for the bottom quark. It employs a robust finite-field reconstruction workflow within FiniteFlow, leveraging IBP reductions to master integrals expressed in pentagon-function bases, and implements the results in a publicly available C++ library. By detailing UV/IR renormalization, $\ ext{$\mu$-dependence}$, and extensive validation (Ward identities, OpenLoops cross-checks), the work provides numerically stable, ready-to-use hard functions for NNLO QCD predictions and massification-based 4FS approximations. The analytic structure is clarified through cancellations among pentagon-function letters, enabling efficient evaluation and paving the way for precise $b\bar{b}H$ phenomenology at the LHC and related processes.

Abstract

We present the double-virtual amplitudes contributing to the production of a Higgs boson in association with a $b\bar{b}$ pair at the Large Hadron Collider. We perform the computation within the five-flavour scheme, which employs massless bottom quarks and finite bottom-Yukawa coupling, taking into account all the colour structures. We derive the analytic form of the helicity amplitudes through finite-field reconstruction techniques. The analytic expressions have been implemented in a public C++ library, and we demonstrate that evaluations are sufficiently stable and efficient for use in phenomenological studies.

Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC

TL;DR

The paper delivers the first full-colour analytic two-loop five-point amplitudes for in the five-flavour scheme, including complete colour structures and finite Yukawa coupling for the bottom quark. It employs a robust finite-field reconstruction workflow within FiniteFlow, leveraging IBP reductions to master integrals expressed in pentagon-function bases, and implements the results in a publicly available C++ library. By detailing UV/IR renormalization, \mu, and extensive validation (Ward identities, OpenLoops cross-checks), the work provides numerically stable, ready-to-use hard functions for NNLO QCD predictions and massification-based 4FS approximations. The analytic structure is clarified through cancellations among pentagon-function letters, enabling efficient evaluation and paving the way for precise phenomenology at the LHC and related processes.

Abstract

We present the double-virtual amplitudes contributing to the production of a Higgs boson in association with a pair at the Large Hadron Collider. We perform the computation within the five-flavour scheme, which employs massless bottom quarks and finite bottom-Yukawa coupling, taking into account all the colour structures. We derive the analytic form of the helicity amplitudes through finite-field reconstruction techniques. The analytic expressions have been implemented in a public C++ library, and we demonstrate that evaluations are sufficiently stable and efficient for use in phenomenological studies.

Paper Structure

This paper contains 17 sections, 56 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Structure of the amplitudes
  3. Kinematics
  4. Two-quark two-gluon scattering channel
  5. Four-quark scattering channel
  6. Renormalisation and infrared subtraction
  7. $\mu$ dependence of the finite remainder
  8. Amplitude calculation
  9. Tensor decomposition and helicity amplitudes
  10. Analytic computation with finite-field arithmetic
  11. Validation
  12. Results
  13. Benchmark numerical evaluation of the hard functions
  14. Numerical implementation
  15. Analytic properties
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Representative two-loop Feynman diagrams appearing in the $0\to\bar{b}bggH$ scattering process, together with the partial amplitudes associated with each of the diagrams. Dashed lines denote the Higgs boson, while solid lines denote massless quarks (red for the bottom quark).
  • Figure 2: Representative two-loop Feynman diagrams appearing in the $0\to\bar{b}b\bar{q}qH$ scattering process, together with the partial amplitudes they contribute to. The notation is the same as in Fig. \ref{['fig:bbggH2L']}.
  • Figure 3: Ordered two-loop integral families entering the IBP reduction: three pentagon-boxes (PB), three hexagon-triangles (HT), three hexagon-boxes (HB), and two double-pentagons (DP).
  • Figure 4: Estimated numerical precision of $\mathcal{H}^{(2)}(p,\mu)$ evaluated for phase-space points arising during Monte Carlo integration given in terms of number of estimated correct digits. The results are separated according to the channels ($q\bar{q}/b\bar{b}/bb/\bar{b}\bar{b}$ and $gg$). Only points that fail the $n^{\text{cor}} > 4$ requirement are re-evaluated at the next higher numerical precision implemented. The estimated evaluation times contain the re-evaluations and are also averaged over the higher precision evaluations, and therefore represent the realistic times per phase-space point during Monte Carlo integration. In total 100000 points have been evaluated for each channel.